Impurity screening by defects in (1+1)$d$ quantum critical systems

TL;DR

The paper shows that impurities in $(1+1)$-D quantum critical states described by CFTs can be screened not only by chiral primary fields but also by topological defect lines (TDLs), when the impurity shares quantum numbers with gapless bulk modes. It recasts impurities as edge modes of symmetry-protected topological (SPT) states and uses SPT-CFT stacking and TDL fusion to generate symmetry-enriched, potentially non-Cardy boundary conditions; this is illustrated with an $SU(3)_{1}$ CFT at a spin-1 chain critical point, where a non-Cardy boundary state screened by a TDL is identified and validated via spectra and the Affleck-Ludwig entropy. The authors corroborate their BCFT predictions with extensive DMRG studies of the ULS spin-1 chain with spin-$ frac{1}{2}$ impurities, observing spectra and scaling consistent with the theory, and they extract the AL entropy $g_{\mathcal{D}_{1/2}} = 3^{1/4}$ from integrable overlaps. They also examine a Spin$(5)_1$ CFT realized by a spin-2 chain, where conformal embedding yields multiple TDLs and potential boundary-state realizations, but finite-size effects and possible RG-flow crossovers complicate the comparison. Overall, the work broadens BCFT by showing non-Cardy boundary conditions arising from TDLs and SPT stacking, with implications for impurity physics in higher-dimensional and more general CFT contexts.

Abstract

We propose a novel mechanism of impurity screening in (1+1)$d$ quantum critical states described by conformal field theories (CFTs). An impurity can be screened if it has the same quantum numbers as some gapless degrees of freedom of the CFT. The common source of these degrees of freedom is the chiral primary fields of the CFT, but we uncover that topological defect lines of the CFT may also take this role. Theoretical analysis relies on the insight that the impurities can be interpreted as edge modes of certain symmetry-protected topological (SPT) states. By stacking a SPT state with a CFT, one or two interfaces on which the SPT edge modes reside are created. If screening occurs due to topological defect lines, a symmetry-enriched CFT with exotic boundary states are obtained. The boundary conditions that appear in these cases are difficult to achieve using previously known methods. As a concrete example, we consider a spin-1 chain whose bulk is described by the SU(3)$_{1}$ CFT and edges are coupled to spin-1/2 impurities. We demonstrate that both the low-energy eigenstates and the extracted Affleck-Ludwig entropy are in excellent agreement with our theoretical predictions.

Figures (7)

• Figure 1: Schematics of our model. (a) The middle part is a spin chain whose low-energy physics is described by a CFT. Its two edges are attached with spin-1/2 impurities. (b) The spin-1/2 impurities in panel (a) can be viewed as the edge states of the AKLT chain.
• Figure 2: Numerical results on the ULS chains with spin-1/2 impurities. (a,b) Energy spectra for the cases with two impurities or one impurity. The ground-state energy is shifted to zero. The total spin of each level is indicated. The averaged spacings $\Delta_{1,2}$ defined in the main text are illustrated. (c,d) Finite-size scaling analysis of the energy spacings $\widetilde{E}_{i} = E_{i}-E_{0}$. (e,f) Finite-size scaling analysis of the energy splittings $\delta_{12},\delta_{34}$ for two impurities and $\delta_{23}$ for one impurity. (g,h) Finite-size scaling analyis of the ratios $R=\Delta_{2}/\Delta_{1}$.
• Figure 3: Finite-size scaling of $\ln |\langle\psi_0(L)|\mathrm{AKLT}(L)\rangle | + \alpha L$ versus $1/\ln L$. The blue open circles represent numerical data obtained using the overlap formula. The results with $L\in [800,1200]$ are fitted by the red dashed line with equation $0.273 - 0.139/\ln L$. For reference, exact value of the AL entropy $\ln g_{\mathcal{D}_{\frac{1}{2}}} \approx 0.275$ is indicated by the purple dot.
• Figure A1: Numerical results on the ULS-AKLT chains. (a,b) Energy spectra for the cases with PBC and OBC. The ground-state energy is shifted to zero. The total spin of each level is indicated. The avergaed spacings defined in the text are illustrated. (c,d) Finite-size scaling analysis of the energy spacings $\widetilde{E}_{i}=E_{i}-E_{0}$. (e,f) Finite-size scaling analysis of the energy splittings $\delta_{12},\delta_{34}$ for PBC and $\delta_{23}$ for OBC. (g,h) Finite-size scaling analyis of the ratios $R=\Delta_{2}/\Delta_{1}$.
• Figure A2: The integral contour $\mathscr{C} = \mathscr{C}_+ + \mathscr{C}_-$ in Eq. \ref{['eq:contour-SM']}, with $\xi \in (0,\frac{1}{2})$.